Posts Tagged ‘polyform covers’

A cover of a set of polyforms is a shape (or set of shapes) into which each member of the set could fit. Mostly I’ve looked at problems involving minimizing the size of a cover. This problem goes the other direction.

A reducible cover is one where a cell can be removed and the remaining figure is still a cover. An interesting problem then is to find an irreducible cover in a single piece that is as large as possible. (Why a single piece? Well, without specifying that, the largest irreducible cover will simply be all of the shapes in the set in separate pieces.) Here’s a (conjectured) maximal irreducible contiguous cover (MICC) of the pentominoes:

The above solution has been on my polyomino cover page for a while. Here are a couple of new results, (still just conjectured since I found them by hand rather than exhaustive computer search, and I am not able as yet to prove they are maximal.)

An MICC (?) of the hexiamonds

An MICC (?) of the pentaedges (shown in two copies for clarity)

Between these solutions, we see some patterns emerging. Certain polyforms are in some sense distinctive: they have features that do not occur in other polyforms in the set. This makes it easy to make a large cover that includes exactly one copy of them. Other polyforms end up serving a connective function. For example, there are quite a few occurrences of the L pentomino in the first figure, so removing a cell will never make the cover cease to include an L. By using a few pentominoes as many times as possible in this connective function, more pentominoes are left over to occur singularly. In some cases multiple polyforms that occur only once are forced to overlap, so we don’t get their full number of cells to add to the cover, but we do get a few. This is shown with the outlined hexiamonds above. In the case of the pentominoes, we have one cell where two T pentominoes overlap; since these are the only two T pentominoes in the figure, the cell can’t be removed from the cover.

Problem #25: Find maximal irriducible contiguous covers of anything and everything! This problem ought to yield interesting results for any kind of polyform you can throw at it.

One final note: It was slightly unfortunate that I chose the word “cover” to represent a concept in polyforms when it already had an unrelated meaning in graph theory; it’s even more problematic now that I’m using graphs themselves as polyforms. It appears that in graph theory, the appropriate term is “common supergraph”. I could use “common superform”, although one problem is that polyforms, unlike graphs, are generally not allowed to be disconnected, and for some problems (though not this one) we want sets of polyforms that aren’t connected to each other. Perhaps “common superformsets” in that case, as ugly as it sounds.

My first particularly interesting and novel discovery in the field of polyforms was what I call the minimal succinct cover problem. The basic minimal cover problem, which I introduced in a previous post, is to find the smallest shape into which a each of the polyforms in a set (e. g. the 12 pentominoes) could fit. A succinct cover is a shape or set of shapes into which each of the polyforms in a set can fit in exactly one way. Several years ago I wrote a program in Python to find minimal succinct polyomino covers, and shortly thereafter I extended it to handle polyhexes. But at the time I left off handling polyiamonds, because of the three regular planar tessellations, (square, hexagonal, and triangular) the triangular one is the trickiest to program.

Still, it was about the lowest hanging fruit around as far as results of mine to extend go, and since I recently blogged the minimal hexiamond cover problem, it seemed only natural to look at minimal succinct polyiamond covers now, so I hacked on my code to add them.

Here’s a minimal succinct cover of the hexiamonds:

Notice that the hexagon must appear as a separate piece in a succinct cover. As a result of its symmetry, whenever you add a cell to it, the resulting heptiamond will fit a couple of hexiamonds in two different ways. (The X pentomino has an analogous position in a succinct pentomino cover.)

And here’s an MSC of the heptiamonds:

(No animation here; my process for making them has a couple of manual steps, which would get tedious for larger sets.)

Notice that there are three hexiamonds in the minimal hexiamond cover, and two heptiamonds in the minimal heptiamond cover, and that all of these have some kind of symmetry. This should not be terribly unexpected; symmetrical polyforms are, in a sense, less flexible to work with because the number of distinct ways you can append a neighboring cell is reduced.

I define a cover of a set of polyforms as a shape or shapes into which each polyform in the set can fit. I’ve written up an exploration of related problems on my recreational mathematics non-blog. Most of these problems use pentominoes or other polyominoes; it lately occurred to me that I had done a disservice to the hexiamonds, which are just as worthy of attention. So here’s a minimal (ten cell) cover of the twelve hexiamonds:

A proof of its minimality is simple. There are two ways that the bar and hexagon hexiamonds can be superimposed with maximal overlap:

Each of these contains nine cells, and neither is a cover of all of the hexiamonds. (The one on the left is missing the butterfly and chevron hexiamonds, the one on the right, just the butterfly. ) Therefore any cover must contain at least ten cells, so the one given above is minimal. (The nomenclature I’m using for individual hexiamonds is given on this MathWorld page.)

In fact, there are five ways to complete a cover by adding a single triangle (marked in blue below) to one of the figures above:

And these are the minimal covers produced:

The animation at the top of this post displays a cycle where a single cell is moved at each step. (I posted about this previously with the pentominoes in a minimal cover.) The hexiamonds have the handy property that their number is exactly twice their size, which leads naturally to the following problem:

Problem #16: Starting with a hexiamond with all cells labeled, find a sequence of single cell moves that cycles through all twelve hexiamonds, returning to the first hexiamond, and that moves each labeled cell exactly twice. Bonus points if the set of cells used is either a minimal cover or a 2×3 parallelogram. Bonus points as well for satisfying conditions like those of problems #10 and #11 in the post linked above. (All cells return to their starting positions, or all cells end up in a cyclic permutation of their starting positions.)

Here’s a problem that I heard from Gordon Hamilton at Gathering for Gardner 9, and tracked down to an article of his in issue #10 of Pi in the Sky, a western Canadian math magazine for high school students and teachers.

A polyomino animal can eat another polyomino animal (his perhaps overly cute term is “polyanimal”) if the second one can be placed inside the first. Find animals of sizes 4, 5, 6, 7, 8, 9, and 10 that can live together peacefully (none can eat any of the others) within a 7×7 square pen.

This is really a satisfying puzzle to solve. Usually in polyform tiling puzzles, you spend a fair amount of time feeling out the territory, learning which pieces like to go in certain places, and which you want to deal with first and which you want to save for the end. But then, the larger part of the solving time is spent in trial and error with various configurations attempted at random until at last you run into a solution.

Here, the whole solving process is learning about the territory of the puzzle, and none of it feels like random crunching. I highly recommend giving it a try, but if you just want to see a solution, mine is here.

Of course, the matter of polyominoes fitting inside other polyominoes is an area that I’ve dealt with in my Polyomino Cover material, which I summarized in my presentation for G4G9. And one of the problems in Hamilton’s Polyanimal problem set is the same as what I’ve called the minimal pentomino cover problem. But most of them are completely different, which only reinforces my belief that this is an area with a lot more waiting to be discovered. (His last problem is “Design a Polyanimal Game.” Now that’s open-ended and provocative!)

What’s the smallest shape into which any of the 12 pentominoes can be placed? I call this old chestnut the “minimal pentomino cover” problem, and I’ve spent a lot of time working on a number of variations on it. For the purpose of introducing and illustrating the basic problem to my dear readers, I wanted to use an animated GIF file showing all of the pentominoes in turn being placed on a minimal cover.

An aesthetically pleasing way to cycle through the pentominoes would be to move one square at a time. This is in fact possible:

A couple of variations on the problem of finding such a cycle suggest themselves:

#9: Minimize the total distance the squares move per cycle. The taxicab metric seems to be more sensible and simpler than Euclidean distances here. I made no attempt to do any minimization in the above solution, so I’m sure there is room for improvement.

#10: If you gave every square in the pentominoes a distinct color, and kept the color the same when a square moved, you could keep track of where the squares end up at the end of a cycle. During the cycle illustrated above, two pairs of squares switch places. Is there a cycle of single-square moves through the pentominoes that ends with each square in the same place it began?

Notice that the central square can never move, because the only pentomino placement without the central square is one of the P pentomino, for which the only valid square movements turn it into a U pentomino. It would need movements to two different pentominoes to be part of a cycle.

For both of the above problems, the other 9 square pentomino cover would also be a valid substrate:

Since this one has no immobile squares, another problem using it may be solvable:

#11: Find a cycle where the permutation of the squares from one cycle to the next is cyclic (in the second sense in the linked article.) That is, successive iterations of the cycle will eventually take each square in a pentomino to all of the other positions in that pentomino.

Some very good news: I’ve been invited to the 9th Gathering for Gardner conference in Atlanta later this month. The Gathering for Gardner is an invitation-only conference held in honor of Martin Gardner, who brought recreational mathematics to a generation through his columns in Scientific American. That generation was not my generation, but it was impossible to miss his imprint on later writers, and I’ve picked up used copies of several of the collections of his columns. A large proportion of the names on the spines on my recreational mathematics bookshelf are represented among the invitees, so this will be really special for me.

I have had a lifelong interest in recreational mathematics, especially polyomino problems. I’ve produced some puzzles in laser-cut plastic, which I sell via this very site. I’ve also dabbled in writing interactive fiction.